The Trefftz method for solving eigenvalue problems

The Methods of External Excitation for Analysis of Arbitrarily-Shaped Hollow Conducting Waveguides

“…The growth of the amplitude of response near the eigenvalue is a sequence of the degeneracy of the matrix of the linear algebraic system which approximates the BVP. From this point of view the presented approach is similar to the one described in [29], where the degeneracy is measured by the infinitesimal values of the minimal eigenvalue of the stiffness matrix of the problem. Recently this technique has been applied for solving problems of free vibrations of beams, membranes and plates [30][31][32].…”

“…We consider TM case (Dirichlet conditions) and set the symmetry conditions ∂w/∂y = 0 along the interval y = 0, 0 ≤ x ≤ 1. This problem as the eigenvalue problem of a cracked beam is studied in detail in [29]. We look for the MFS solution in the form:…”

“…Indeed, the functions of the Kupradze basis become very smooth when the singular point is far from the solution domain. To overcome the difficulties, the Trefftz method (TM) [29] and the method of particular solutions (MPS) [38][39][40] have been developed. These techniques use various particular solutions of the eigenvalue equation which describes the local behaviour of the …”

“…Using the package Mathematica, the first two eigenvalues were calculated in [29] with 13 significant digits. They are shown in the last column of the table.…”

The method of external excitation for solving Laplace singular eigenvalue problems

“…However, this method faces great difficulties when applied to the problems with boundary singularities. To overcome them, the Trefftz method (TM) [9] and the method of particular solutions (MPS) [10][11][12] were developed. These techniques use various particular solutions of the eigenvalue equation which describe the local behaviour of the eigenfunction near the singular points.…”